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Trading & Hedging Local Volatility (The Journal of Financial Engineering)

This note outlines a methodology for hedging and trading index volatilities.

In the bond world, forward rates are the arbitrage-free interest rates at future times that can be locked in by trading bonds today. Similarly, in the world of index options, local volatilities are the arbitrage-free volatilities at future times and market levels that can be locked in by trading options today. The dependence of local volatility on future time and index level is called the local volatility surface, and is the analog of the forward yield curve. In this paper we show how to hedge portfolios of index options against changes in implied volatility by hedging them against changes in future local volatility. This is analogous to hedging bond portfolios against changes in forward rates.

Eurodollar futures on interest rates are the best-suited instrument for forward-rate hedging. Unfortunately, there are no liquid futures on local index volatility. So, we will define a volatility gadget, the volatility analog of a Eurodollar futures contract. A gadget is a small portfolio of standard index options that is sensitive to local index volatility only at a definite future time and index level, and, like a futures contract, has an initial price of zero.

We can create unique volatility gadgets for each future time and index level. By buying or selling suitable quan- tities of gadgets, corresponding to different future times and market levels, we can hedge an index option portfolio against any changes in future local volatility. This proce- dure is theoretically costless. It can help remove unwanted volatility risk, or help acquire desired volatil- ity exposure, over any range of index levels and times where we think future local volatility changes are likely to occur.

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